Hello Higher Dimensional geeks! This is my first post and it is a question.

Is it possible to deform a 3D space (or object, it does not matter) in the absence of 4D?

Here is analogy on -1 D: a piece of paper represents a 2D plane. I can crumble or bend it, only because I am doing it from 3D. But if the 3rd D did not exist, I would not be able to deform the piece of paper that represents a 2D. In absence of a 3rd dimension, there would not be freedom for the 2D plane to bend in that direction. So, the "forces that comprise the 2D" would hold the sheet perfectly flat. It is the presence of a 3rd dimension that allows the deformation of a 2D space (or object).

In other words, what I am trying to prove is that a N+1 dimension is absolutely required in order to deform an N-dimensional object (or space).

Am I wrong? Where?

Thank you for your help

PSI also realize that this is also a question of the nature of deformations. For example, an elastic 2D can be deformed by twisting (say, pulled diagonally), but still remain in 2D constrains. I realize that 3D space can be deformed by compression/extension without leaving the 3D domain. The deformation of the 3D I'm talking about should be different, just like it is plainly seen in a bend of a sheet of paper in 3D. What would be a similar deformation of a 3D object in 4D? Lets say the object is a box, but not a cube, more like a shoebox, so that all sides are different.

Please help me out here.

PPSyou know, while I was typing this, the answer occurred to me: it depends on the rigidity of the object in question. The more rigid the N-object, the harder for it to be deformed in the absence of N+1 dimension.

Interesting thoughts. Thinking about it a piece of paper in a 2D world would be just a line and different letters would have to be differentiated possibly by colour as well along with length I guess. Certainly it would be limiting.So I guess a 3D piece of paper would get more crumply than a 2D piece of paper which can only develop curve/fold crumples.

It appears that I did not formulate my question well, causing you misunderstand it. My appologies

With a piece or paper, I made a simplified analogy (i.e. Flatland), showing that its 2D plane must exist in 3 dimensions in order for it to be able to be distorted, like a crumbled piece of paper in our real world. Which implies that, if a hypothetical universe that contained Flatland did not have a 3rd spatial dimension, the plane of Flatland would always be perfectly... well, flat.

What I am looking for is a topological theorem (if there is such a thing, and I'm assuming it is) which would state that an (N+1)th dimension must exist in order for a N-dimensional.. whatever to be distorted. (the N-dimensional whatever is expressed in a perfect Euclidean N-geometry, so that any deviation from its perfection constitutes the distortion -- simple!).

The problem is plainly obvious for a 2-dimensional world in 3D (our model with a sheet of paper). It is not so obvious for a 3-dimensional world in 4D. I was hoping that people here could point where I could learn more about this topic.

As I was typing the response above, it occurred to me that.. the answer has to do with the surface (i.e. a plane) from which the (n+1)th dimension emerges.

In the case of Flatland, the 3rd dimension emerges orthogonally from its 2D sheet. If I sample 2 points on Flatland's 2D plane, by drawing a line through each point into the 3rd dimension, orthogonal to the plane, and then compare the angle between the resulting 2 lines, then, if the angle is 0, it means that they are parallel to each other and the area between these 2 sampling points is flat. If not, this means that the 2D plane is curved in 3D between these 2 points.

Everything is examined from the POV of a hypothetical 3D Euclidean space. Pretty straightforward.

In 4D, a 3D object has 3 bounding planes, orthogonal to each other in a unique way (XY, XZ and YZ). From each of these 3 planes, the 4th dimension emerges (W), and it is equally orthogonal to all these 3 planes. (I probably am not using the terms right )

Looking at 3D from within 3D, with nothing distorted:

XY ⊥ XZ, XY ⊥YZ , XZ ⊥YZ and if I draw a line a ⊥ XY, line b ⊥ XZ and line c ⊥ YZ,these 3 lines, a b and c, are not parallel to each other. (they are in fact ⊥ to each other).

Now looking at 3D from 4D, no distortions:

again I draw 3 lines orthogonal to XY, XZ and YZ, but this time I draw them into the 4th dimension:

a ⊥ XY, b ⊥ XY, c ⊥ YZin the 4th dimension, these 3 lines, a b and c, are parallel to each other.

In other words, from the point of view of the 4th dimension, the 3 planes XY, XZ and YZ belong to the same 4-plane -?

So, my question essentially is: instead of me re-inventing this bicycle, I'd like to avail myself of an already existing, properly formulated topological theorem. It gotta be out there. What's it called and where can I find it?

I'm not sure I understand your question. In your last 3D->4D example, there is actually an entire plane that's orthogonal to, say, the XY plane in 4D, so there are an infinitude of lines you may draw that are orthogonal to the XY plane, and any other 3D plane. So given your coordinate planes (I believe that's the term) XY, YZ, and XZ, you could draw orthogonal lines that make any arbitrary angle with each other (up to a certain extent).

But I'm not sure I understand what this has to do with your original question of n dimensions implying n+1 dimensions.

In physics, general relativity implies the curvature of (3+1)-dimensional space-time, but this curvature need not be happening in an additional dimension. For all we know, 3D space could be distorted within the 3D Euclidean geometry itself. For example, take the case of 2D space: imagine a piece of cloth stitched in such a way that the strings cross each other to form a grid. This grid then serves as a reference for our coordinate system, such that lines and curves are shaped relative to this grid. Now imagine stretching the cloth such that the distance between threads is no longer regular, but some parts are more stretched in one direction and more compressed in another direction. This can happen without the cloth leaving the plane at all. (You can see this for yourself by taking an old piece of cloth in which the threads aren't as tightly woven anymore, and holding it down on a flat surface while stretching one of its corners with your finger: the threads will stretch/compress w.r.t. to each other without leaving the plane if you don't pull at it too hard.) If we were to still use the strings as reference for our coordinate system, we find that this 2D space is now distorted, but the fabric of space itself (excuse the pun) is just as "flat" as before.

So in the case of 3D space, just because there are distortions in space doesn't necessarily require the existence of higher dimensions (though they can). It may be that the "threads in the 3D cloth" just happen to be compressed/stretched, without ever requiring an additional dimension.

Thank you quickfur for your reply. Sorry I was not clear -- it's cause my understanding is evolving. I am new to topology.

I understand your example with threads in a piece of fabric and agree that 3D space can be distorted within 3D, without leaving its constrains. But. Think about it: if that were true with gravity in our world, then space would not appear to us perfectly flat, especially here on Earth. Spacetime is heavily curved around the planet, all around us. Yet we do not perceive this curvature in the refraction of light. (I'm aware that there is a lensing effect around stars, that's not what I'm talking about here.) What I'm saying is that space all around us on the planet is condensed and curved, yet we do not see it in all sorts of halos, mirages and lensings. This is possible ONLY if the distortion is in fact in the 4th spatial dimension, while the 3D is practically perfect.

The analogy here is with a thin 2D sheet of Flatland that is gently curved in 3D, appearing perfectly orderly and regular for the Flatlanders.

So, when I analyzed the situation in regard to curvature in (n+1) dimension, I saw that what curves is the surface plane from which the (n+1)th dimension arises. And to determine if it's curved, for 2D seen from 3D, I need to make orthogonal lines from 2 arbitrary points on the plane and see if they are parallel or not.

When I went from 3D to 4D, I noticed that the 3 planes that bind a cube from one side, all belong to the same plane from which the 4th dimension emerges (<-- as seen from the 4th D point of view).

I hope this is right -?

I want to learn to visualize this 4D thing well. Actually, I need only to see a 3D surface from the 4th D perspective, which is much easier. That's how I realized that the 3 faces of a cube lie on the same plane in 4D. - is that right?

Please talk to me and reassure me that I am doing at least something right. Point out where I am wrong. thanks

When one says that space is curved, it is not bent in some higher space. All space, including euclidean, is curved. So if one supposes that 3d is curved in a 4d holding space, then the 4d holding space is curved in a 5d space, and so forth.

Curvature actually has to do with the difference of lengths in an arc of a circle. Near large masses, the lengths of arcs of equal angle are slightly longer then on the side opposite the mass. The tension of space will work towards where there is more length, so the tug of space itself will cause a mass to move towards another mass. When a ray of light falls across the circle's centre, the entry and exit points divide the length of arc equally, not the angle. So near a mass, where there is more length, the light appears to bend (ie has an angle less than 180 degrees), towards of the mass.

In the geometry of hyperbolic space, even euclidean space is horibly curved, even though its curvature is numerically zero. Straignt lines have the same curvature as all-space.

The dream you dream alone is only a dream
the dream we dream together is reality.

4Dspace wrote:Thank you quickfur for your reply. Sorry I was not clear -- it's cause my understanding is evolving. I am new to topology.

I understand your example with threads in a piece of fabric and agree that 3D space can be distorted within 3D, without leaving its constrains. But. Think about it: if that were true with gravity in our world, then space would not appear to us perfectly flat, especially here on Earth. Spacetime is heavily curved around the planet, all around us. Yet we do not perceive this curvature in the refraction of light. (I'm aware that there is a lensing effect around stars, that's not what I'm talking about here.) What I'm saying is that space all around us on the planet is condensed and curved, yet we do not see it in all sorts of halos, mirages and lensings. This is possible ONLY if the distortion is in fact in the 4th spatial dimension, while the 3D is practically perfect.

I don't follow your reasoning. A "fabric distortion", if we may call it that, does not necessarily mean halos and mirages, etc., if the distortion is only slight. Even a 4D-style distortion would cause all sorts of halos and distortions (as you point out with the gravitational lensing example, except on a more extreme scale) if the gravitational constant were much larger. A fabric that is only slightly distorted would show almost no visible effects, until you extend it to a very much larger piece of fabric, in which case the accumulated effect of the slight distortions will start showing up in a visible way.

The analogy here is with a thin 2D sheet of Flatland that is gently curved in 3D, appearing perfectly orderly and regular for the Flatlanders.

So, when I analyzed the situation in regard to curvature in (n+1) dimension, I saw that what curves is the surface plane from which the (n+1)th dimension arises. And to determine if it's curved, for 2D seen from 3D, I need to make orthogonal lines from 2 arbitrary points on the plane and see if they are parallel or not.

Correct. What you just described is the idea behind a "manifold" - locally Euclidean, but globally not necessarily so. So on the small scale, it looks like Euclidean space, but slightly distorted so that over the large scale, it actually forms a curved surface.

But anyway, with regard to your orthogonal lines, note that those lines would lie outside the 2D space, which is a bit troublesome if you're confined to working within the 2-manifold itself (since you would have no access to the 3rd dimension where the lines extend). A more convenient means of measuring curvature, incidentally, is to observe how the parallel postulate fails to hold within the 2D space itself. So hyperbolic spaces will see parallel lines eventually diverge, whereas elliptic spaces will see parallel lines eventually intersect. These are the two idealized cases, of course, but one can easily have position-dependent violations of the parallel postulate, in which case you'll have other kinds of global shapes for the space.

One advantage of this approach is that you don't need to add the assumption that (n+1)-dimensional space exists; one can imagine a space "fabric" of uneven "spatial density", like the fabric distortion I mentioned, that can also give rise to the same kind of effects (violations of the parallel postulate) without needing n+1 dimensions. So by using this approach, it can apply to both situations, whereas conclusions drawn by assuming n+1 dimensions may no longer apply in the "fabric distortion" cases.

When I went from 3D to 4D, I noticed that the 3 planes that bind a cube from one side, all belong to the same plane from which the 4th dimension emerges (<-- as seen from the 4th D point of view).

It would help to use unambiguous terminology, so that we're all on the same page. Keep in mind that in 4D, you have both 2-planes (planes spanned by two linearly independent vectors), which do not divide 4D space, and 3-planes, which do divide 4D space. One needs to distinguish between them in order to not get into trouble.

A cube (3D cube) in 4D unambiguously defines a hyperplane, and there is exactly one vector orthogonal to this hyperplane. However, each face of the cube only defines a 2-plane, and in 4D, there is another 2-plane that is orthogonal to this 2-plane. So a 2-plane is not sufficient to define an orthogonal vector in 4D.

I hope this is right -?

I want to learn to visualize this 4D thing well. Actually, I need only to see a 3D surface from the 4th D perspective, which is much easier. That's how I realized that the 3 faces of a cube lie on the same plane in 4D. - is that right? [...]

This is correct if by "plane" you mean "hyperplane", that is, the 3-plane defined by the cube. It is not to be confused with the 2-planes that each face of the cube lies in.

As for 4D visualization, here's my shameless plug: http://eusebeia.dyndns.org/4d/vis/vis I use 4d->3D projections as my primary method of 4D visualization, because I found that it is most intuitive to me. YMMV.

quickfur, thank you so much for your explanation. And for your website, I am going to explore it now.

Regarding light distortions: when light passes through a lens or a prism, or simply uneven window glass (like they used to make in medieval times, 'cause they did not have the technology), the image such light delivers gets distorted. If spacetime is curved by gravity all around us, why don't we see similar distortions all around? <-- that's my line of reasoning.

Wendy above hints that... well, the way I understand it is.. sort of Lorenz' transformation that compensates for the distortion, evening it out in the end -? something like this. (what she says is way above my head...)

Also, Wendy said,

wendy wrote:When one says that space is curved, it is not bent in some higher space. All space, including euclidean, is curved. So if one supposes that 3d is curved in a 4d holding space, then the 4d holding space is curved in a 5d space, and so forth.

Here, as a novice, I have the following problems:

1. The way I understand the definition of Euclidean space is that it is "perfect" (perfectly flat, that is). I understand, that a subspace may appear Euclidean, while being distorted in a higher dimensional space that contains it (please forgive my lack of knowledge of proper terms).

2. I do not see how it follows that because a subspace is curved (say, 3d), it implies that the holding space (4d) has to be curved too, and so on. This so on part, I don't get at all. Because, ultimately, I examine a structure from the point of view of Absolute Space (perfectly Euclidean and having as many dimensions as I need to fully see the structure). I realize that Absolute Space is an abstraction that exists only in my head, but it must be perfect Euclidean space, because a curvature or distortion in the structure I am examining can be judged only against this perfection. -? So, in the end, in this nested Russian doll model, the largest doll gotta be the perfect Euclidean space of enough dimensions to hold the structure under examination. That's how my mind operates. Where I am wrong?

wait... do you mean that.. you can arbitrarily select a dimension in a given structure and start "counting dimensions" from it -? ..meaning, which subset you call 2d and which is holding it 3D and so on? Yes, in this case, where all dimensions are "equal in their rights" so to speak, yes, I understand the point (-? I hope ) But. What if dimensions are NOT equal? What if dimensions arise as a result of some physical process? (say, when space folds.. then the resulting dimensions are NOT the same in.. their qualities-? as the ones that "gave birth" to them).

I'm gonna spend some time exploring your 4D visualization site -- thank you so much for it. Please give me feed back on my line of reasoning, especially whatever strikes you as wrong.

4Dspace wrote:quickfur, thank you so much for your explanation. And for your website, I am going to explore it now.

Regarding light distortions: when light passes through a lens or a prism, or simply uneven window glass (like they used to make in medieval times, 'cause they did not have the technology), the image such light delivers gets distorted. If spacetime is curved by gravity all around us, why don't we see similar distortions all around? <-- that's my line of reasoning.[...]

We do. Gravitational lensing is the prime example. I suppose your objection is that we don't see it more often -- but that's just a matter of degree. Most distortions are so tiny we don't notice it -- doesn't mean they aren't there. It has been shown, for example, that large nearby mountains have enough gravity to pull very long pendulums sideways. So the distortion is definitely there. It's just that when you couple light, which moves at extremely great speeds, and gravity, which is very weak as far as physical forces go, the effects are barely noticeable until you start talking about truly massive amounts, like entire galaxy clusters and millions of light years of distance. The spacetime around us does have all kinds of distortions, but they are slight enough that we don't readily notice them (e.g. the mountain bending a pendulum).

But this is really beside the point. The point is that space distortions don't have to require an additional dimension for them to be real. If you are confined to the surface of a piece of fabric, it's not so easy to tell, from your confined POV, whether the distortion is caused by curvature in 3D, or just stretching/compression within the fabric itself. In fact, it may not be possible to tell in some cases, without going outside of the fabric itself. You simply don't have enough information within the fabric itself to be able to prove conclusively that an additional dimension must be involved.

Anyway, I gotta run, will answer the rest of your post later. Hope this helps.

Thank you quickfur for your reply. I'm just coming from you site. Thank you for it! It's beautiful. I get such a pleasant feeling when I look at perfect shapes in beautiful colors. Your 4d-cube is the best I've ever seen.

quickfur wrote:We do. Gravitational lensing is the prime example. I suppose your objection is that we don't see it more often -- but that's just a matter of degree. Most distortions are so tiny we don't notice it -- doesn't mean they aren't there. It has been shown, for example, that large nearby mountains have enough gravity to pull very long pendulums sideways. So the distortion is definitely there. It's just that when you couple light, which moves at extremely great speeds, and gravity, which is very weak as far as physical forces go, the effects are barely noticeable until you start talking about truly massive amounts, like entire galaxy clusters and millions of light years of distance. The spacetime around us does have all kinds of distortions, but they are slight enough that we don't readily notice them (e.g. the mountain bending a pendulum).

Neah. Light passing through the atmosphere of Earth, bouncing off surfaces of familiar objects before hitting our retinas, should be greatly distorted. Yet we see nothing. Here on Earth. Gravitational lensing around a galaxy or a star is an extreme case. We should see something on Earth with our own eyes. But we don't. Why?

You're saying this is because the distortion is so small, we don't notice it. But when glass has a minutest imperfection, we notice right away. We can tell very well when light is distorted ever so slightly. So, what's the difference?

I say, the difference between these two cases is in the 3d subspace (here I am referring to the fact that there are 4 3d subspaces in 4D). Gravity lives in 4D proper; i.e. it needs all 4 dimensions. Light (EM radiation) lives in 3d subspace of 4D. Prob'bly I am not using the right terms

quickfur wrote:But this is really beside the point. The point is that space distortions don't have to require an additional dimension for them to be real. If you are confined to the surface of a piece of fabric, it's not so easy to tell, from your confined POV, whether the distortion is caused by curvature in 3D, or just stretching/compression within the fabric itself. In fact, it may not be possible to tell in some cases, without going outside of the fabric itself. You simply don't have enough information within the fabric itself to be able to prove conclusively that an additional dimension must be involved.

The point is that there can be different kinds of distortions. In our 2D analogy with a piece of fabric, yes it can be distorted in 2D (its threads having kinks), remaining flat in 3D. But it can also be curved in 3D, looking pretty orderly in 2D (its threads perfectly aligned). There is a difference. I want to know how to tell this difference.

Translating the above to this 2D analogy: light bent by gravity == curvature in 3D while the threads of the piece of fabric have no kinks. Light bent by a defect in glass == the threads that make up the fabric have a kink that does not leave the plane of the fabric. Nothing changes in 3D.

Pleas help me To hell with gravity. Forget it. The question is: How to tell if a 3D structure is distorted in 4D as opposed to the same 3D it is in?

Actually, I just remembered, from computer graphics courses (yeah, me too, except that I ended up working with databases and as a sys analyst). To depict 3D we used 4x4 matrices, and that's because... something to do with the fact that you need n+1 dimension to fully describe n-dimensional object.

I think this has to do with.. an n-D object, no matter how distorted can always be contained in a Euclidean (n+1)D. There gotta be a theorem that states this -?

In matter of various kinds, the speed of light is the square root of the products of the electric and magnetic constants, ie c = sqrt(ce cm). The light we see follows a path of shortest time, the solution being identical to the usual run/swim problem. For something like light in water, we have ce as something like 553 million feet per second. The magnetic constant is very close to the speed of light, 983 million feet per second. We find then the speed of light in water is something like 740 million feet per second. Light hitting water is thus bent, on the fastest time, where light in air travels at 4, and light in water travels at 3, ie a refractive index of 1.333 (740 million = 983 million / 1.333).

However, the curvature in space is not due to this, and by no where as large. Typically, you are looking at feet per light-years here.

One should note that the hyperbolic geometry has a curvature less than euclidean space, and so a space of perfect euclidean curvature would be a curved thing in that kind of space.

Curvature is something that one does not have to suppose a larger space to do. It is partly suffice to construct lines perpendicular to a straight line. One then measures the length at some distance from the straight line. This will be longer, equal or shorter, as the space is negatively, zero or positively curved. If the case is that space is negatively curved, a euclidean construction would be as bent as a sphere. In fact, for these spaces, if one is significantly smaller than the radius of curvature (a measure which connects d(length)/length), then space will appear to be euclidean. Such is true, for example, on the surface of the earth, where one does not have to reckon curvature for the construction of a house, but does for large countries.

The dream you dream alone is only a dream
the dream we dream together is reality.

4Dspace wrote:[...] Light passing through the atmosphere of Earth, bouncing off surfaces of familiar objects before hitting our retinas, should be greatly distorted. Yet we see nothing. Here on Earth. Gravitational lensing around a galaxy or a star is an extreme case. We should see something on Earth with our own eyes. But we don't. Why?

You're saying this is because the distortion is so small, we don't notice it. But when glass has a minutest imperfection, we notice right away. We can tell very well when light is distorted ever so slightly. So, what's the difference?

The difference is that the "minutest imperfection" to us is actually a gigantic mountain of faulty area at the atomic level, consisting of billions, nay, trillions, nay, quadrillions, nay, something on the order of 10^20 (that's 1 with 20 zeroes following it) atoms in size. That's when we macroscopic beings start noticing the effects of it.

Change just a few atoms in the glass, say on the order of 20 atoms or so? Completely undetectable, even to the finest microscopes. Definitely undetectable to the naked eye. You need a scanning electron microscope to even begin to notice a flaw of that size. So your analogy is flawed. It takes a pretty large distortion (large from the atomic point of view) in the glass before we start noticing the effects of it at the macroscopic level.

I say, the difference between these two cases is in the 3d subspace (here I am referring to the fact that there are 4 3d subspaces in 4D). Gravity lives in 4D proper; i.e. it needs all 4 dimensions. Light (EM radiation) lives in 3d subspace of 4D. Prob'bly I am not using the right terms

Well, that's just getting into technicalities. If you really want to be technical, you can draw a distinction between a "smooth" distortion (say the density of a piece of glass varies ever so slightly from 0.9 to 1.1 -- you'd see a gradual light distortion in this case) and a "discontinuous" distortion (there is a clear line or area in the glass where the density suddenly jumps from 0.9 to 1.1, say). Notice that I deliberately used density as an example of how a space might be distorted without requiring additional dimensions. Both examples can be recast in terms of an additional dimension: in the smooth case, the surface is curving smoothly into the extra dimension, and in the discontinuous case, the surface is "folded" along a ridge (like the edge of a polyhedron, if you like concrete examples) so that the transition over the ridge is abrupt.

So you see, spatial distortion can be explained either way. There is really no way to tell the difference unless you have actual access to the additional dimension.

[...]The point is that there can be different kinds of distortions. In our 2D analogy with a piece of fabric, yes it can be distorted in 2D (its threads having kinks), remaining flat in 3D. But it can also be curved in 3D, looking pretty orderly in 2D (its threads perfectly aligned). There is a difference. I want to know how to tell this difference.

You can't, unless you can access the additional dimension. Just because a certain kind of fabric (or glass, if you want to use the density example) doesn't exhibit a certain kind of distortion does not tell you anything about whether there is an additional dimension involved. The kinds of distortions possible is governed by the nature of the forces that hold it together. A fabric made of nylon threads will admit different types of distortions than a fabric made of stiff metal wires. It doesn't tell you anything about additional dimensions, unless you can measure that additional dimension directly. Which we can't in our case, because we're confined within 3D space, so there's no way to measure anything in the 4th direction, assuming there is one.

[...] To hell with gravity. Forget it. The question is: How to tell if a 3D structure is distorted in 4D as opposed to the same 3D it is in?

Unfortunately, you can't. Not unless you already have direct access to said 4D.

4Dspace wrote:Actually, I just remembered, from computer graphics courses (yeah, me too, except that I ended up working with databases and as a sys analyst). To depict 3D we used 4x4 matrices, and that's because... something to do with the fact that you need n+1 dimension to fully describe n-dimensional object.

That's inaccurate. The reason 4x4 matrices are used for 3D modelling is because the use of homogenous coordinates allows you to represent certain non-linear transformations using a higher-dimensional linear transformation (the prime examples being affine translation and projection). This is a mathematical device to simplify calculations, not an inherent property of 3D space itself.

I think this has to do with.. an n-D object, no matter how distorted can always be contained in a Euclidean (n+1)D. There gotta be a theorem that states this -?

This is not necessarily true. The Real Projective Plane, for example, is a 2D object that is distorted in such a way that it cannot be embedded in 3D without self-intersection. But it can, in 4D. The Klein bottle is another example. In both cases, you need two additional dimensions to have a faithful embedding of the object into Euclidean space.

But again, just because some shape can be embedded in n+1 dimensions, doesn't mean that it is. That's a distinction I think you're missing here. Yes, general relativity can be explained by curvature in an additional dimension. But there's no proof that this is actually the case. The real cause of the curvature may be something else altogether.

quickfur wrote: you need two additional dimensions to have a faithful embedding of the object into Euclidean space.

Awesome! Where do I read up on this? How this theorem is called. I wuv you!

So, it's +2, uh? Thank you thank you thank you.

quickfur wrote:spatial distortion can be explained either way. There is really no way to tell the difference unless you have actual access to the additional dimension.

Thank you very much! This was important for me to know. now I do

quickfur wrote:The difference is that the "minutest imperfection" to us is actually a gigantic mountain of faulty area at the atomic level, consisting of billions, nay, trillions, nay, quadrillions, nay, something on the order of 10^20 (that's 1 with 20 zeroes following it) atoms in size. That's when we macroscopic beings start noticing the effects of it.

You're saying that a spot with some messed up atoms on a piece of glass makes a greater distortion in otherwise straight ray of light than the dent the whole planet Earth makes in space? Here we differ. The distortion of space around the planet is so great that it requires phenomenal >11 km/s escape velocity to break through.

In fact, I'm now curious to know the distortion in spacetime at about 1 to 1000 meters from the surface of the planet. How would I go about calculating it?

__________________________

And I have another question regarding the geometry/topology. I'm looking for a theorem that states that a symmetrical and highly ordered 3d structure in 4D is more rigid and stable than a 4d structure in 4D. I'm not sure how "stability" is determined. This gotta go in the same vein as such things as triangles being rigid in 2D or knots can be tied in 3D but not in 4D -? but I am not sure.

wendy wrote:However, the curvature in space is not due to this, and by no where as large. Typically, you are looking at feet per light-years here.

feet per light-years. uh? Could it be that I am way-way off visualizing it?

wendy wrote:One should note that the hyperbolic geometry has a curvature less than euclidean space, and so a space of perfect euclidean curvature would be a curved thing in that kind of space.

Curvature is something that one does not have to suppose a larger space to do. It is partly suffice to construct lines perpendicular to a straight line. One then measures the length at some distance from the straight line. This will be longer, equal or shorter, as the space is negatively, zero or positively curved. If the case is that space is negatively curved, a euclidean construction would be as bent as a sphere. In fact, for these spaces, if one is significantly smaller than the radius of curvature (a measure which connects d(length)/length), then space will appear to be euclidean. Such is true, for example, on the surface of the earth, where one does not have to reckon curvature for the construction of a house, but does for large countries.

These are very interesting thoughts, wendy. Thank you. I'll go meditate on them.

quickfur wrote: you need two additional dimensions to have a faithful embedding of the object into Euclidean space.

Awesome! Where do I read up on this? How this theorem is called. I wuv you!

There is no such theorem that I'm aware of. There is no upper limit to the number of additional dimensions you need.

So, it's +2, uh? Thank you thank you thank you.

It's +2 only in this particular case (of the real projection plane and the Klein bottle). Other objects may require more. One can, for example, twist a 2D sheet in such a way that it requires at least 5 dimensions to represent without self-intersection (just cut out some arbitrary knotted 2D manifold from 5D space, for example). You can potentially go a lot higher than that, too (cut a 2D manifold out of, say, a 100D fabric -- in the general case, you will require about 100D to have a faithful representation of it).

[...]

quickfur wrote:The difference is that the "minutest imperfection" to us is actually a gigantic mountain of faulty area at the atomic level, consisting of billions, nay, trillions, nay, quadrillions, nay, something on the order of 10^20 (that's 1 with 20 zeroes following it) atoms in size. That's when we macroscopic beings start noticing the effects of it.

You're saying that a spot with some messed up atoms on a piece of glass makes a greater distortion in otherwise straight ray of light than the dent the whole planet Earth makes in space? Here we differ. The distortion of space around the planet is so great that it requires phenomenal >11 km/s escape velocity to break through.

It's all a difference of scale. What appears phenomenal to us is nothing from an astronomical POV. For example, the Andromeda Galaxy is moving towards the Milky Way at a rate of up to 140 km/s. That's an order of magnitude faster than the escape velocity of Earth. But on an astronomical scale? It's barely budging, and will take another 4.5 billion years before the two galaxies collide, which is a sizeable fraction of the age of the universe, if current estimates are to be believed.

So saying that the distortion of space is "great" isn't exactly very informative. How great is great? What is great to us is not very great in the grand scheme of things. And besides, it's missing the essence of the matter, which is that spatial distortion can be equally understood as curvature in an additional dimension or as internal, inherent curvature in the space itself. The relative scale of things is irrelevant to this issue, really, since it differs just in what constant factors you put into the model. Use a larger constant and get a larger effect, but it doesn't change the nature of the model.

[...] And I have another question regarding the geometry/topology. I'm looking for a theorem that states that a symmetrical and highly ordered 3d structure in 4D is more rigid and stable than a 4d structure in 4D. I'm not sure how "stability" is determined. This gotta go in the same vein as such things as triangles being rigid in 2D or knots can be tied in 3D but not in 4D -? but I am not sure. [...]

Well, if your definition of stability is unclear, then I'm also unclear how to answer your question.

But it's not true that knots can be tied in 3D but not in 4D... it's true only if you're talking about tying knots with ropes. In 4D, ropes can't be knotted (except trivially -- you just pull them and they become undone). But 2D sheets can. The Klein bottle, for example, is an example of a knotted sheet. The real projective plane is another example. Sheets can't be knotted in 3D (the Klein bottle can't be embedded in 3D without self-intersection, for one thing), but ropes can. In general, in n dimensions, knots are possible with (n-2)-dimensional objects. Any more, and you can't tie them to begin with, any less, and only trivial knots can be made.

As for rigidity, any n-dimensional object made of rigid edges will be rigid in n dimensions. There's no special relationship with specific dimensions. In a 2D world where gravity pulls towards the center of a circle, a two-legged construct is maximally stable. In 3D, a tripod is maximally stable. In 4D, you need a 4-legged construct that has its feet at the vertices of a tetrahedron. And so on. In 5D you have a 5-legged thing that points to the corners of a 5-cell, etc.. In general, in n dimensions you need (n-1) legs pointing at the vertices of an (n-1)-simplex. Perhaps this is what you had in mind?

But it's the same light that passes through distortions, the same visible range, the same frequencies. It's the same scale for the main participant of the event: the light.

Today I spent some time googling trying to find out what's the value of spacetime curvature at ~1m off the surface of the Earth. After so many years, you'd expect that someone has already calculated this -? But I could not find it

quickfur wrote: There's no special relationship with specific dimensions. ... In general, in n dimensions you need (n-1) legs pointing at the vertices of an (n-1)-simplex. Perhaps this is what you had in mind?

Well, I tell you what I have in mind. I'm trying to deconstruct space. The one we live in. To me it's a fascinating subject and I hope you find it interesting too. I have a hunch that space is actually at least 4D. In fact, the rigidity of the 3D electromagnetic structure requires that an additional spatial dimension existed (for nuclei to move somewhere unobstructed). So, pursuing this line of reasoning, I am looking for some rules of topology that would make a 3D rigid structure naturally emerge out of a 4D structure in 4D, that is trying to assume the lowest energy state.

The most rigid and stable structure is actually 2D. The problem with it is that it is not bounded. => it needs to curve. This leads us to 4D <-- because the "real space" does not grow by drawing an orthogonal vector from a plane, it folds: 2d x 2d = 4D = 3D + 1D. This separation of 4D on 3D + 1D is what we got. 3D is occupied by the EM rigid structure and 1 empty D is where nuclei live, integrated into the 3D of EM via electron clouds.

So, what I'm looking is some clues, hidden in topology, as to why 3d structure in 4D precipitated out of a 4d structure in 4D. Any ideas?

Clue: all the structure "wants" is to assume the lowest energy state, to be as regularly distributed as possible. It can be modeled as nodes connected by edges. A node only "knows" what its 2 neighbors are doing and they are either pulling or pushing. The structure overall "wants" to equalize this pull-push among all of its nodes (yeah, it vibrates). That's the principle that drives the organization of this structure.

But it's the same light that passes through distortions, the same visible range, the same frequencies. It's the same scale for the main participant of the event: the light.

You missed my point. What I was saying is that inherent space curvature can produce the same observed effects as curvature into an additional dimension. The whole scale thing was in response to your objections about how distortions in cloth fabric wouldn't produce the same effects as curvature in an additional dimension. I was just saying that if the distortions were small, you wouldn't be able to tell the difference. The analogy is not perfect; I was only using cloth as an example of something that can show curvature-like effects without being actually curved into an additional dimension. Obviously space isn't literally a piece of cloth with a grid of lines running through it. My point was that space itself can have certain constraints that makes it behave in a way that exhibits curvature, but without actually being a curved manifold in n+1 space.

Today I spent some time googling trying to find out what's the value of spacetime curvature at ~1m off the surface of the Earth. After so many years, you'd expect that someone has already calculated this -? But I could not find it

Isn't it just a matter of plugging in Earth's mass into Einstein's equations for general relativity? (That's an understatement, though. I realize how immensely complex Einstein's equations are once you plug actual figures into it. There's a lot of stuff hidden behind those innocent-looking symbols.)

[...]Well, I tell you what I have in mind. I'm trying to deconstruct space. The one we live in. To me it's a fascinating subject and I hope you find it interesting too. I have a hunch that space is actually at least 4D. In fact, the rigidity of the 3D electromagnetic structure requires that an additional spatial dimension existed (for nuclei to move somewhere unobstructed). So, pursuing this line of reasoning, I am looking for some rules of topology that would make a 3D rigid structure naturally emerge out of a 4D structure in 4D, that is trying to assume the lowest energy state.

Ahhh. Now things are a bit more clear. I'm not so sure about your reasoning here -- what do you mean by "nuclei to move somewhere unobstructed"? What obstruction is there?

Also, it sounds like you're going to need a lot more than just topology here, since you're working from electromagnetism. Merely dealing with abstract mathematical spaces here isn't going to get you very far. You have to take into account how electromagnetism works, and then tie that in to the dimensionality of the space concerned. That's way beyond my depth, I'm afraid.

The most rigid and stable structure is actually 2D. The problem with it is that it is not bounded. => it needs to curve. This leads us to 4D <-- because the "real space" does not grow by drawing an orthogonal vector from a plane, it folds: 2d x 2d = 4D = 3D + 1D. This separation of 4D on 3D + 1D is what we got. 3D is occupied by the EM rigid structure and 1 empty D is where nuclei live, integrated into the 3D of EM via electron clouds.

You've lost me there. What do you mean by 2D being "rigid"? And "not bounded"? Bounded by what? You have some unstated assumptions here that I don't know about, and I'm having trouble following your line of thought. And I'd be careful about leaping to conclusions about 4D being 2D x 2D -- that depends on what you mean by "x", and what you're trying to accomplish. In the context of physics, which seems to be what you're concerned with here, there is no reason (that I know of, anyway) to imagine that the space-time coordinates of some event have some kind of special decomposition into 2+2 dimensions. So I don't know where you got your "folding" from (and I'm not sure I understand what you mean by folding here).

So, what I'm looking is some clues, hidden in topology, as to why 3d structure in 4D precipitated out of a 4d structure in 4D. Any ideas?

Clue: all the structure "wants" is to assume the lowest energy state, to be as regularly distributed as possible. It can be modeled as nodes connected by edges. A node only "knows" what its 2 neighbors are doing and they are either pulling or pushing. The structure overall "wants" to equalize this pull-push among all of its nodes (yeah, it vibrates). That's the principle that drives the organization of this structure.

OK, so any physical system seeks to be in the minimal energy state. So you're trying to find a reason for EM to be (3+1)D instead of something else by an energy minimization argument? You might want to look into quantum electrodynamics -- there is a discussion of it somewhere that I read before (sorry don't remember where it was) that gives a dimension-independent description of it. Search for gauge symmetry. I think that would be a good starting point; the usual QED descriptions pre-assume 3+1D so aren't helpful if that's what you're trying to arrive at. Starting with gauge symmetry may give you some hints as to why it comes out with 3+1D instead of anything else.

Sorry, quickfur, you did not understand me. Forget physics. Think space. Think there is nothing in the world but the structure of space. When Universe is born, the first expression of its energy is the geometry of space. Space wants to be an even structure. And it quickly becomes. When it is perfectly balanced, that's what is called empty space. Each and every imperfection in its structure is something in space. And so, there is absolutely nothing in the world but topology describing the dynamic structure of space.

There is no physical forces in conventional understanding. Instead, there are distortions in the structure of space.

And so, you don't need to concern yourself with physics. The question to you is, why would a 4d structure in 4D precipitate into a 3d structure in 4D? Everything else is superfluous. I am not dealing with anything but structure of space itself. Nuclei and other trinkets are irrelevant here.

4Dspace wrote:Sorry, quickfur, you did not understand me. Forget physics. Think space. Think there is nothing in the world but the structure of space. When Universe is born, the first expression of its energy is the geometry of space. Space wants to be an even structure. And it quickly becomes. When it is perfectly balanced, that's what is called empty space. Each and every imperfection in its structure is something in space. And so, there is absolutely nothing in the world but topology describing the dynamic structure of space.

There is no physical forces in conventional understanding. Instead, there are distortions in the structure of space.

And so, you don't need to concern yourself with physics. The question to you is, why would a 4d structure in 4D precipitate into a 3d structure in 4D? Everything else is superfluous. I am not dealing with anything but structure of space itself. Nuclei and other trinkets are irrelevant here.

Where did the 4D structure come from in the first place? And what do you mean by "precipitate"? Apart from physics, you just have mathematical spaces, and mathematical spaces are just spaces... they don't go around "precipitating". I don't understand what you're talking about.

quickfur wrote:Where did the 4D structure come from in the first place?

What does it matter? This is a given, like in chess: you have a position and the goal: white mate black in 3 moves. You don't ask how the position came about.

quickfur wrote: And what do you mean by "precipitate"?

By this I mean that, given a 4d structure in 4D (initially, the structure occupies all of the 4D = yes, initially the structure also defines the space), in order to get to the lowest energy state (it vibrates), it precipitates into a 3d structure.

The way I sort of see it, it collapses into 3d, because this configuration is more conducive to preservation of its energy. The 4 dimensions, from which it started though, remain. The 4th dimension becomes very different from the other 3. First, it appears "empty" in comparison to the other 3 where all the energies continue to run. This dimension sort of "helps" to absorb some of the "rattle" of the other 3. I know I sound phenomenally primitive, but... hey I don't know the terms yet. Where online I could read on the topology and functional spaces?

PSIt just occurred to me that, having proved this for 4D->3D, one can make an (n+1) case and prove it for all other dimensions ( I still can't believe that a theorem stating just that does not already exist -- I just can't adequately describe it! help me please )... then, having proved it for (nD -> (n-1)D) --> Lim(->3) [-? forgot the notation, 3 is the limit to which it converges], this will explain "Why 3D". In English, it would sound, like... the 3D configuration is where its energy is most preserved. (meaning that, whenever local pressures let go of it, the structure tries to get there) That's all. Simple.

quickfur wrote:Where did the 4D structure come from in the first place?

What does it matter? This is a given, like in chess: you have a position and the goal: white mate black in 3 moves. You don't ask how the position came about.

It matters because you appear to be making an assumption about the initial state of space which is non-obvious, so it seems reasonable to ask what justification you have for such an assumption.

quickfur wrote: And what do you mean by "precipitate"?

By this I mean that, given a 4d structure in 4D (initially, the structure occupies all of the 4D = yes, initially the structure also defines the space), in order to get to the lowest energy state (it vibrates), it precipitates into a 3d structure.

You keep saying the physics don't matter, yet here you talk about energy and vibrations, which don't exist in a mathematical space. I'm confused.

The way I sort of see it, it collapses into 3d, because this configuration is more conducive to preservation of its energy. The 4 dimensions, from which it started though, remain. The 4th dimension becomes very different from the other 3. First, it appears "empty" in comparison to the other 3 where all the energies continue to run. This dimension sort of "helps" to absorb some of the "rattle" of the other 3. I know I sound phenomenally primitive, but... hey I don't know the terms yet. Where online I could read on the topology and functional spaces?

What do you mean by "functional space"? I hope you realize that there is such a term in mathematics, and it probably doesn't mean what you think it means.

It seems that the kind of spaces you're talking about are physical spaces, not mathematical ones. A mathematical space doesn't "precipitate" or "collapse" or "vibrate"; it is just a mathematical construct that obeys the rules set forth in its definition. Sure, a mathematical space may be a good description of the shape of some physical space, but it doesn't have physical laws to govern its change over time as a physical space does. It's important not to conflate the two, lest you get the wrong idea that mathematical spaces can somehow change or move around, etc.. Change comes from the application of physical laws. (And physical laws don't necessarily have to talk about matter; there are some physics models, for example, where matter is not an actual "thing" but merely a vibrational mode of space. In such a case, the way that space changes is governed by some postulated physical laws.) Energy is a physical concept; mathematical objects don't possess energy in and of themselves.

Now, since we're talking about physical spaces (you can think of it as mathematical space + a set of postulated physical laws that govern how that space evolves over time), we need to ask what are the physical laws that govern that space. For example, I can start with Euclidean 2D space, and postulate that its curvature increases by 1 every year. So I get a model of 2D space that is initially Euclidean, and over time becomes more and more hyperbolic. Of course, such a space doesn't reflect anything in real-life, but perhaps somebody might find such a model interesting for study. My point is, before you can go about making conclusions about physical spaces, you need to be sure that your postulated physical laws actually reflect reality, or at least, is a reasonable approximation thereof. Otherwise you have an interesting model, but it doesn't tell you anything about the real world.

Of course, one may also start with a model that is self-consistent, and work out its consequences to see if it matches up with reality. If it does, then the model may reflect the underlying nature of reality -- after rigorous testing with physical evidence, of course. This could also be thought of as trying to find a model that, when worked out, matches reality. Generally, this is harder -- much harder -- than starting with known physical laws.

It seems that you're trying to do the latter, by finding some kind of topological property of 3D or 4D that may cause space to prefer that number of dimensions. To do this, first you need to be clear about what it is, that causes a particular topological property to be preferred. Mathematics isn't going to give you an answer: a topological property is just a topological property; there is no special preference for a particular property, mathematically speaking. The preference only comes when you have physical laws -- for example, a certain kind of topology may have a particular energy associated with it, and a low-energy state is presumably preferred. Of course, you'll also have to define how exactly one assigns energy values to a particular topology, since mathematical objects, as I stated, don't inherently have an "energy". And one always has to find some justification for why a particular set of laws were chosen (why that topology has a particular energy, or why energy is calculated in that way), and whether this reflects the real-world, or is merely an interesting model.

Since you haven't stated how exactly topology relates to energy, I really don't know how to help you here, particularly since you appear to be describing 4D space in a way that is different from generally-accepted Minkowskian 4D geometry of general relativity, so I don't know what assumptions I can make or what I can go on.

This page lists some features peculiar to each dimension, some topological, some speculative. While I don't see any direct correlation with your idea of why 3D is preferred, perhaps you can find something there that might be helpful.

Thank you, quickfur. I can't describe how very much I value your input

So, I was thinking about this plane that is the result of cross product in 4D. Does it have chirality? Cause the structure I have in mind is dynamic. Lines have directions and planes have chirality.

How does vector/matrix notation differs from a static plane to plane with chirality? It was a long way for me, besides, in CGI courses i never had to deal with chirality So, I don't know. But I know that the result of a cross product in 3D is a vector and not just a line. So, what about this plane? Which way it turns? And if it does not, can chirality be simply assigned to it?